We have the following quote from Eichler: "There are five elementary arithmetical operations: addition, subtraction, multiplication, division, and… modular forms."

Why did Eichler consider modular forms "elementary arithmetical operations"? Not to mention that subtraction and division are just addition and multiplication with inverse elements. Can anybody shed some light on his train of thought?

4 Answers
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Modular forms are of course not "elementary", but they are a basic operation of arithmetic in the following sense: Modular forms can be used to generate sequences of integers that are most naturally defined and studied in the context of modular forms, and can also be used to prove arithmetic identities that would be significantly harder without modular forms. Furthermore, I think modular forms are unique in this respect of being a way of producing integers/rational numbers using means that cannot be reduced to the four operations of addition, subtraction, multiplication, and division.

The classic example of this is the Ramanujan tau function; tau(n) is most naturally defined as the coefficient of q^n in the unique weight 12 cusp form Delta for SL_2(Z) with leading term q. One can come up with a number of formulas for tau(n) in terms of things that can be defined using only the four standard operations of arithmetic (some of them can be found on the mathworld page above), but the easiest way to prove that these formulas all define the same function uses modular forms. Likewise, most of the other properties of tau listed on that Mathworld page are most easily understood in terms of propreties of the associated modular form Delta.

My interpretation of the quote is different. (Granted, whether Eichler really said such a thing is a different question). It should refer to solving the algebraic equations, to the legacy of Abel and Galois and to Kronecker's Jugendtraum! To the book of Taniyama and Shimura, whereby certain abelian equations are solved by adjoining the moduli of a relevant abelian variety. In this sense, from the point of view of exact solutions to algebraic equations, the modular functions such as $j$ are indeed a natural, inevitable complement to the basic arithmetic operations $+, \times, \sqrt[n]{\cdot}$. This falls into the rubric of explicit class field theory. For a connection to modular forms such as $\Delta(\tau)$, see Ribet's converse to Herbrandt's theorem; a great introduction to this circle of ideas is Mazur's article, "How can we construct abelian Galois extensions of basic number fields?" (Bull. AMS, vol. 48, no. 2, pp. 155-209).

One question is whether Eichler really did say it. One of the nice things about Google searches these days is that they let you trace the etymology of words, phrases, and aphorisms. I see a chain of references to this quote associated with Andrew Wiles and Fermat's Last Theorem. The chain seems to begin with a BBC interview in 1996 or 1997 in which Wiles says, "There's a saying attributed to Eichler that there are five fundamental operations of arithmetic: addition, subtraction, multiplication, division, and modular forms." On the other hand, this recent paper by Edixhoven, van der Geer, and Moonen calls the quote apocryphal.

I can't trace what happened before that interview. Since the documented history of this quip currently begins there, we can start with what Wiles himself meant. The interviewer wanted to know what a modular form is, but that's not easy to explain on television. Obviously it was a facetious (and witty) answer. It was also part of a general point in the interview that modular forms are widely studied and seem fundamental.

in p-adic Hodge theory, categories (semistable, crystalline, de-Rham, Hodge-Tate) of p-adic representations of Galois groups over local fields are proved to be equivalent to linear algebra categories (filtered modules with compatible actions). But for Galois representation of global fields, no such linear categories seem to exist, since the Galois group is more complicated, but then there are modularity results. Saying that these Galois representations correspond to modular forms (the most basic level). Of course, then according to Langlands Program, there are generalizations to automorphic forms.
So, in some sense, modular forms are a "global patching" of all local linear data.